"No party has claimed responsibility," a statement said, adding that authorities were continuing to investigate the incident and "take the necessary measures to preserve Egypt's interests and national security".
The drone reportedly hit a US-owned floating storage and regasification vessel, Energos Winter, sparking a blaze that spread to a Greek-owned LNG tanker, GasLog Salem.
Verified videos posted on social media showed clouds of black smoke rising from the port and tugboats hosing down the ships with water.
The fires were extinguished and no casualties were reported by Egyptian authorities, who said the port was continuing to operate at full capacity on Thursday.
Energos Winter and GasLog Salem left the port and moved offshore following the incident, according to ship tracking data.It is not clear where the drone was launched or who was behind the attack.
However, Iranian forces and Iranian-backed armed groups have carried out similar attacks on tankers in the Gulf and Red Sea as well as energy infrastructure in several Arab states during the US and Israel's war with Iran, disrupting global trade and causing oil prices to soar.
US President Donald Trump told reporters on Wednesday that he had been briefed on what happened in Damietta and implied that Iran or its proxies were responsible."It's a little more of the same. But it's going to be straightening out," he said.
The Damietta attack came hours after the US and Saudi Arabia carried out joint air strikes on Iran-backed armed groups in Iraq that they said had attacked Saudi oil facilities with drones 30 times in recent days at the direction of Iran's Islamic Revolutionary Guard Corps (IRGC).
Iraq's paramilitary Popular Mobilisation Forces (PMF), which is dominated by Iran-backed Shia militias, said at least 20 of its members were killed in the strikes, which they called an "extremely serious escalation".
Iran's foreign ministry also condemned the strikes and warned of "dangerous consequences".
On Wednesday night, the US carried out what it described as a "heavy wave of strikes" on Iran in response to an attempted "surprise" missile attack on US military facilities in Jordan.